(C2xC8):6F5 - GroupNames

G = (C₂×C₈)⋊₆F₅ order 320 = 2⁶·5

4^th semidirect product of C₂×C₈ and F₅ acting via F₅/D₅=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₂×C₈)⋊₆F₅, (C₂×C₄₀)⋊₆C₄, (C₈×D₅)⋊₇C₄, D₅.D₈⋊₅C₂, C₄₀⋊C₄⋊₆C₂, C₈.₂₇(C₂×F₅), C₄₀.₂₄(C₂×C₄), (C₄×D₅).₈₉D₄, C₄.₃₀(C₄⋊F₅), C₂₀.₃₀(C₄⋊C₄), (C₄×D₅).₃₂Q₈, D₅.₁(C₄○D₈), D₁₀.₃₁(C₂×D₄), C₅⋊(C₂³.₂₅D₄), C₄⋊F₅.₁₇C₂², C₂².₅(C₄⋊F₅), D₁₀.₂₉(C₄⋊C₄), C₄.₃₇(C₂²×F₅), C₂₀.₇₇(C₂²×C₄), Dic₅.₁₅(C₂×Q₈), (C₂×Dic₅).₃₅Q₈, (C₄×D₅).₇₇C₂³, (C₈×D₅).₅₄C₂², (C₂²×D₅).₉₉D₄, Dic₅.₃₀(C₄⋊C₄), D₁₀.C₂³.₁₁C₂, (D₅×C₂×C₈).₂₂C₂, (C₂×C₅⋊₂C₈)⋊₂₀C₄, C₂.₁₆(C₂×C₄⋊F₅), C₁₀.₁₃(C₂×C₄⋊C₄), C₅⋊₂C₈.₄₇(C₂×C₄), (C₄×D₅).₈₆(C₂×C₄), (C₂×C₄).₁₃₈(C₂×F₅), (C₂×C₁₀).₂₁(C₄⋊C₄), (C₂×C₂₀).₁₄₇(C₂×C₄), (C₂×C₄×D₅).₄₀₃C₂², SmallGroup(320,1059)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂₀ — (C₂×C₈)⋊₆F₅

Chief series

C₁ — C₅ — C₁₀ — D₁₀ — C₄×D₅ — C₄⋊F₅ — D₁₀.C₂³ — (C₂×C₈)⋊₆F₅

Lower central

C₅ — C₁₀ — C₂₀ — (C₂×C₈)⋊₆F₅

Upper central

C₁ — C₄ — C₂×C₄ — C₂×C₈

Generators and relations for (C₂×C₈)⋊₆F₅
G = < a,b,c,d | a²=b⁸=c⁵=d⁴=1, ab=ba, ac=ca, dad^-1=ab⁴, bc=cb, dbd^-1=b³, dcd^-1=c³ >

Subgroups: 442 in 114 conjugacy classes, 50 normal (38 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₅, C₈, C₈, C₂×C₄, C₂×C₄, C₂³, D₅, D₅, C₁₀, C₁₀, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂²×C₄, Dic₅, C₂₀, F₅, D₁₀, D₁₀, C₂×C₁₀, C₄.Q₈, C₂.D₈, C₄²⋊C₂, C₂²×C₈, C₅⋊₂C₈, C₄₀, C₄×D₅, C₂×Dic₅, C₂×C₂₀, C₂×F₅, C₂²×D₅, C₂³.₂₅D₄, C₈×D₅, C₂×C₅⋊₂C₈, C₂×C₄₀, C₄×F₅, C₄⋊F₅, C₂²⋊F₅, C₂×C₄×D₅, C₄₀⋊C₄, D₅.D₈, D₅×C₂×C₈, D₁₀.C₂³, (C₂×C₈)⋊₆F₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₂³, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, F₅, C₂×C₄⋊C₄, C₄○D₈, C₂×F₅, C₂³.₂₅D₄, C₄⋊F₅, C₂²×F₅, C₂×C₄⋊F₅, (C₂×C₈)⋊₆F₅

Smallest permutation representation of (C₂×C₈)⋊₆F₅
►On 80 points

Generators in S₈₀

(1 21)(2 22)(3 23)(4 24)(5 17)(6 18)(7 19)(8 20)(9 69)(10 70)(11 71)(12 72)(13 65)(14 66)(15 67)(16 68)(25 55)(26 56)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 63)(34 64)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)(41 77)(42 78)(43 79)(44 80)(45 73)(46 74)(47 75)(48 76)

(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)

(1 50 42 71 59)(2 51 43 72 60)(3 52 44 65 61)(4 53 45 66 62)(5 54 46 67 63)(6 55 47 68 64)(7 56 48 69 57)(8 49 41 70 58)(9 35 19 26 76)(10 36 20 27 77)(11 37 21 28 78)(12 38 22 29 79)(13 39 23 30 80)(14 40 24 31 73)(15 33 17 32 74)(16 34 18 25 75)

(1 19 5 23)(2 22 6 18)(3 17 7 21)(4 20 8 24)(9 54 80 59)(10 49 73 62)(11 52 74 57)(12 55 75 60)(13 50 76 63)(14 53 77 58)(15 56 78 61)(16 51 79 64)(25 43 38 68)(26 46 39 71)(27 41 40 66)(28 44 33 69)(29 47 34 72)(30 42 35 67)(31 45 36 70)(32 48 37 65)

G:=sub<Sym(80)| (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,69)(10,70)(11,71)(12,72)(13,65)(14,66)(15,67)(16,68)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,77)(42,78)(43,79)(44,80)(45,73)(46,74)(47,75)(48,76), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80), (1,50,42,71,59)(2,51,43,72,60)(3,52,44,65,61)(4,53,45,66,62)(5,54,46,67,63)(6,55,47,68,64)(7,56,48,69,57)(8,49,41,70,58)(9,35,19,26,76)(10,36,20,27,77)(11,37,21,28,78)(12,38,22,29,79)(13,39,23,30,80)(14,40,24,31,73)(15,33,17,32,74)(16,34,18,25,75), (1,19,5,23)(2,22,6,18)(3,17,7,21)(4,20,8,24)(9,54,80,59)(10,49,73,62)(11,52,74,57)(12,55,75,60)(13,50,76,63)(14,53,77,58)(15,56,78,61)(16,51,79,64)(25,43,38,68)(26,46,39,71)(27,41,40,66)(28,44,33,69)(29,47,34,72)(30,42,35,67)(31,45,36,70)(32,48,37,65)>;

G:=Group( (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,69)(10,70)(11,71)(12,72)(13,65)(14,66)(15,67)(16,68)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,77)(42,78)(43,79)(44,80)(45,73)(46,74)(47,75)(48,76), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80), (1,50,42,71,59)(2,51,43,72,60)(3,52,44,65,61)(4,53,45,66,62)(5,54,46,67,63)(6,55,47,68,64)(7,56,48,69,57)(8,49,41,70,58)(9,35,19,26,76)(10,36,20,27,77)(11,37,21,28,78)(12,38,22,29,79)(13,39,23,30,80)(14,40,24,31,73)(15,33,17,32,74)(16,34,18,25,75), (1,19,5,23)(2,22,6,18)(3,17,7,21)(4,20,8,24)(9,54,80,59)(10,49,73,62)(11,52,74,57)(12,55,75,60)(13,50,76,63)(14,53,77,58)(15,56,78,61)(16,51,79,64)(25,43,38,68)(26,46,39,71)(27,41,40,66)(28,44,33,69)(29,47,34,72)(30,42,35,67)(31,45,36,70)(32,48,37,65) );

G=PermutationGroup([[(1,21),(2,22),(3,23),(4,24),(5,17),(6,18),(7,19),(8,20),(9,69),(10,70),(11,71),(12,72),(13,65),(14,66),(15,67),(16,68),(25,55),(26,56),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(33,63),(34,64),(35,57),(36,58),(37,59),(38,60),(39,61),(40,62),(41,77),(42,78),(43,79),(44,80),(45,73),(46,74),(47,75),(48,76)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80)], [(1,50,42,71,59),(2,51,43,72,60),(3,52,44,65,61),(4,53,45,66,62),(5,54,46,67,63),(6,55,47,68,64),(7,56,48,69,57),(8,49,41,70,58),(9,35,19,26,76),(10,36,20,27,77),(11,37,21,28,78),(12,38,22,29,79),(13,39,23,30,80),(14,40,24,31,73),(15,33,17,32,74),(16,34,18,25,75)], [(1,19,5,23),(2,22,6,18),(3,17,7,21),(4,20,8,24),(9,54,80,59),(10,49,73,62),(11,52,74,57),(12,55,75,60),(13,50,76,63),(14,53,77,58),(15,56,78,61),(16,51,79,64),(25,43,38,68),(26,46,39,71),(27,41,40,66),(28,44,33,69),(29,47,34,72),(30,42,35,67),(31,45,36,70),(32,48,37,65)]])

44 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	···	4N	5	8A	8B	8C	8D	8E	8F	8G	8H	10A	10B	10C	20A	20B	20C	20D	40A	···	40H
order	1	2	2	2	2	2	4	4	4	4	4	4	4	···	4	5	8	8	8	8	8	8	8	8	10	10	10	20	20	20	20	40	···	40
size	1	1	2	5	5	10	1	1	2	5	5	10	20	···	20	4	2	2	2	2	10	10	10	10	4	4	4	4	4	4	4	4	···	4

44 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	4	4	4	4	4	4
type	+	+	+	+	+				+	-	-	+		+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	C₄	D₄	Q₈	Q₈	D₄	C₄○D₈	F₅	C₂×F₅	C₂×F₅	C₄⋊F₅	C₄⋊F₅	(C₂×C₈)⋊₆F₅
kernel	(C₂×C₈)⋊₆F₅	C₄₀⋊C₄	D₅.D₈	D₅×C₂×C₈	D₁₀.C₂³	C₈×D₅	C₂×C₅⋊₂C₈	C₂×C₄₀	C₄×D₅	C₄×D₅	C₂×Dic₅	C₂²×D₅	D₅	C₂×C₈	C₈	C₂×C₄	C₄	C₂²	C₁
# reps	1	2	2	1	2	4	2	2	1	1	1	1	8	1	2	1	2	2	8

Matrix representation of (C₂×C₈)⋊₆F₅ ►in GL₆(𝔽₄₁)

32	23	0	0	0	0
9	9	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	40	0
0	0	0	0	0	40

11	11	0	0	0	0
15	0	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	40	40	40	40
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0

40	0	0	0	0	0
1	1	0	0	0	0
0	0	40	0	0	0
0	0	0	0	0	40
0	0	0	40	0	0
0	0	1	1	1	1

G:=sub<GL(6,GF(41))| [32,9,0,0,0,0,23,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[11,15,0,0,0,0,11,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,1,0,0,40,0,0,0],[40,1,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,1,0,0,0,0,40,1,0,0,0,0,0,1,0,0,0,40,0,1] >;

(C₂×C₈)⋊₆F₅ in GAP, Magma, Sage, TeX

(C_2\times C_8)\rtimes_6F_5

% in TeX

G:=Group("(C2xC8):6F5");

// GroupNames label

G:=SmallGroup(320,1059);

// by ID

G=gap.SmallGroup(320,1059);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,100,1684,102,6278,1595]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^5=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^4,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=c^3>;

// generators/relations

G = (C2×C8)⋊6F5 order 320 = 26·5

4th semidirect product of C2×C8 and F5 acting via F5/D5=C2

G = (C₂×C₈)⋊₆F₅ order 320 = 2⁶·5

4^th semidirect product of C₂×C₈ and F₅ acting via F₅/D₅=C₂